RU2329511C2 - Force-sensitive resonator - Google Patents

Abstract

FIELD: mechanics.

SUBSTANCE: invention relates to measurements of mechanical force and its derivatives, i.e. moment of force, pressure, weight, deformations, linear and angular accelerations. The force-sensitive resonator is made up of one core or a system of two cores with the combined ends forming a two-branch tonometer. At the central part of the core and at one of the ends elastic hinges are arranged. The core part between elastic hinges features higher bending resistance compared to that of the other part.

EFFECT: higher relative sensitivity, high-quality resonator and higher performance.

3 dwg

Description

The invention relates to the field of measurements of mechanical force and related quantities: moment of force, pressure, mass, deformation, linear and angular accelerations. A known piezoresonance sensor (see application No. 200/324393 of 08/04/2003, published in BI No. 4 of 02/10/2005), which is the closest in technical essence to the claimed device and is taken as a prototype.

A resonator with a bending waveform is made in the form of at least one rod, the ends of which are rigidly connected to the areas of application of the measured force. The disadvantage of the prototype is the limited ability to obtain high sensitivity with a minimum size of the resonator due to a decrease in its quality factor.

The technical task to be solved is the creation of a device with higher sensitivity and greater quality factor.

The technical result is to reduce the critical strength of the rod and its resonant frequency by changing the shape of the bending vibrations of the resonator. The technical result is achieved by the fact that the power-sensitive resonator is made of at least one rod, the ends of which are rigidly connected to the areas of application of the measured force. New is the fact that in the middle part and at one of the ends of the rod elastic joints are formed, while the width of the section of the rod enclosed between the elastic joints is greater than the width of the rest of the section of the rod.

On figa) shows the shape of the resonator made of a plate in the form of a single rod; on figb) shows a resonator made in the form of a system of two rods with combined ends (two-fork tuning fork). On figa) presents a kinematic diagram of a rod loaded with a longitudinal force P corresponding to the rod resonator of the prototype; on figb) presents a kinematic diagram of the resonator according to the invention, loaded with a longitudinal force P. On figa) presents an equivalent mechanical diagram of the resonator according to the invention, and fig.3b) its electrical equivalent circuit according to the first system of electrical analogies.

The resonator in figa), b) is formed of the following elements seamlessly connected: sections of the rods 1, 2 with the formation of an elastic joint 3 between them; the ends of the rods are connected to the sections 4, 5 of the application of the measured force F, while at the places of transition of the sections of the rods 2 to the section 4 of the application of the measured force, elastic hinges 6 are formed, and the section 5 is rigidly connected to the other end of the rod in the section 1. The resonator in the form of a two-branch tuning fork, the combined ends of the rods of sections 1 are connected to the measuring force application section 5 using the foot 7, the width of the rods in section 2 is more important than that of sections 1. On the surface of the rods using metal methods ation (spraying, plating deposition and the like) applied electromechanical transducer electrodes (not shown in Figure 1). Depending on the physical properties of the material used in the resonators, the electromechanical transducers for excitation and registration of its vibrations can be of a piezoelectric, magnetoelectric or electromagnetic type, their operating principle is described in the technical literature, in particular in the books of P.V. Novitsky and others. "Digital devices with frequency sensors "," Energy "1970, pp. 136-141; or G.A. Filatov et al. "Small-sized low-frequency mechanical filters", "Communication" 1974, pp. 33-74.

The impedance of the electromechanical converter on the electrical side at the frequency of the resonance of the resonator takes extreme values relative to values outside the resonance frequencies, this is used to build generators of electrical signals with a generation frequency equal to the frequency of the mechanical resonance. The device operates as follows. The process of resonant bending vibrations of rod resonators, as well as other types of resonators, is accompanied by the exchange of potential and kinetic energies between two reactive elements of the system with distributed parameters:

equivalent elasticity and equivalent mass; part of the energy is lost by friction in the material of the rod during its bending deformation and in the places of attachment of the ends of the rod (see the equivalent circuit of Fig. 3a). The use of a system of two rods with united ends performing antiphase bending vibrations as a resonator allows one to reduce friction losses at the ends of the joints, and to increase the strength of the resonator (thereby making the resonator in the form of a two-branch tuning fork).

When a measured force is applied to the ends of the rod (sections 4, 5 of FIG. 1), the frequency of the mechanical resonance changes, and accordingly, the oscillation frequency of the generator, to which the mechanical resonator is connected through its electromechanical converter, changes. The value of the frequency of the mechanical resonance f of the rod resonator with a bending waveform for the first mode when a longitudinal force acts on it is determined by the following expression:

where P is the longitudinal force;

f ₀ is the value of the resonant frequency at P = 0;

B is the reciprocal of the critical force P _{cr of the} resonator rod (longitudinal force at which the rod loses stability).

при значениях ВР в подкоренном члене выражения (1) не более 0,1 в первом приближении может быть представлена линейным членом разложения в степенной ряд функции

The conversion function of a power-sensitive resonator, expressed through the relative change in its resonant frequency (relative deviation)

when the values of BP in the root term of expression (1) are not more than 0.1 in a first approximation, can be represented by the linear term of the expansion in the power series of the function

In the conversion function (2), design parameters (linear dimensions of the resonator rod, physical characteristics of the material, conditions for fastening the ends) are implicit in the expression for the quantity B (the reciprocal of the critical force P _cr ). Using the well-known general expression for critical strength:

We get the transformation function (2) in the form:

where l is the length of the resonator rod;

μ is the coefficient of reduction of the length of the rod, is determined by the conditions for fastening its ends;

E is the elastic modulus of the rod material;

j is the moment of inertia of the rod.

For resonators with a rectangular cross section, which have received the widest application, the expression for the moment of inertia j has the form:

where b is the width of the rod;

h is the thickness of the rod (bending vibrations in the plane l, h).

Then, taking into account expression (4) and Hooke's law: σ = Еε (σ are the mechanical stresses in the rod under the action of the measured force P; ε is the relative deformation of the rod), the transformation function will have the form:

Where

The transformation function (6) can also be represented as a dependence on the relative strain ε:

Expressions (6) and (7) for the relative deviation of the resonant frequency of the power-sensitive resonators allow comparative evaluations of various designs of rod resonators that differ in size, conditions for attaching the ends of the rods, physical characteristics of the materials (elastic modulus E, ultimate stresses σ _pr ). Depending on the mode of application: the regime of a given force or the regime of a given deformation, formula (6) or (7) is used, respectively.

In formulas (6) and (7), the factors in front of σ and ε are the coefficients of relative sensitivity to mechanical stress C _σ and relative strain C _{ε of the} resonator, due to the action of the longitudinal measured force or a given deformation, respectively.

As can be seen from expressions (8) and (9), the sensitivity depends on the square of the ratio of the length of the rod of the resonator to its thickness l / h, the coefficient of reduction of the length μ, determined by the conditions of attachment of the ends of the rod and the elastic modulus of the material of the rod for a given force regime. The possibility of obtaining maximum sensitivity is limited by the maximum value of the l / h ratio, which for the rod resonator should be less than 300, because for large values, the rod resonator passes into the class of string resonators requiring the initial tension of the string. Another limitation of increasing sensitivity by varying the dimensions of the resonator — decreasing the thickness h is a decrease in the Q factor of the resonator and a decrease in the efficiency of the electromechanical transducer (for resonators with a piezoelectric transducer).

Another design parameter that affects the sensitivity of the resonator is the coefficient of reduction of length μ. In the resonator prototype with a rigid attachment of the ends to the areas of application of the measured force, the value of the coefficient μ is 0.5; when replacing the rigid attachment of one of the ends with a hinge coefficient μ increases to 0.7, which allows you to double the relative sensitivity with other equal design parameters.

The implementation of another elastic hinge in the area between the pivotally fixed end and the other rigidly fixed end (see fig.2b) while maintaining the length of the portion of the rod between the rigidly fixed end and the hinge in the middle part equal to the length of the rod of the resonator prototype (figa), allows you to further increase the coefficient of reduction of length μ (to reduce the critical force P _cr ). In the book of V.I. Feodosiev "Resistance of materials", ed. "Science", Moscow, 1974, pp. 466-447 presents a solution to the problem of determining the critical force for the kinematic scheme of Fig.2b. The result is a transcendental equation:

wherein the through k ² denotes the ratio

Given the general expression P _cr (3), the formula for the reduction coefficient μ will have the form:

The effectiveness of the proposed technical solution, in terms of increasing sensitivity, can be demonstrated by the example of one of the possible versions of the resonator, in which the length a of the rigid part of the rod (connecting rod) located between two hinges is equal to the length of the rest of the rod l located between the elastic hinge in the middle part and a rigidly fixed end; those.

The solution of equation (12) is the value kl≈1,165. For the value kl≈1,165, the coefficient of reduction of the length will be equal to: μ≈2.70.

The resonator of the prototype with a rigid attachment of the ends with a uniform cross section along the length of the rod, as indicated above, the coefficient of reduction of length is 0.5.

The gain in sensitivity, determined by the relative deviation, of the proposed technical solution relative to the prototype can be expressed in terms of the ratio of the sensitivity coefficients of the resonators (if the general conditions are equal):

Or

The criterion for equality of conditions when comparing the proposed technical solution with the prototype is:

- equal dimensions along the length of the rods between the points of attachment of the ends;

- equality of the thickness of the rods;

- the equality of the elastic modulus E of the materials of the rods. Assigning index 1 to the characteristics of the proposed resonator and index 2 to the resonator characteristics of the prototype, taking into account the equality of the general conditions of expression (13), (14) will take the form:

wherein

where L is the length of the rod of the resonator prototype between the points of connection of its ends with the areas of application of the measured force;

and - the size of the hard portion 2 of the resonator, according to the proposed technical solution, located between the elastic joints 3, 6 (see figure 1).

значение μ₁, как указывалось выше, равно 2,7 (коэффициент μ₂ прототипа равен 0,5), значение отношения (15) равно:For an embodiment of the proposed resonator with the ratio

the value of μ ₁ , as mentioned above, is 2.7 (the coefficient μ _{2 of the} prototype is 0.5), the value of the ratio (15) is:

Thus, the implementation of elastic hinges at the junction of one of the ends of the resonator rod with the site of application of the measured force and in the middle part allows, ceteris paribus, several times to increase the conversion coefficient (sensitivity). The above qualitative analysis was verified by computer simulation using the finite element method. The simulation results confirmed a significant gain in the sensitivity of the proposed technical solution compared to the prototype.

The implementation of elastic hinges 3, 6 on the rods of the resonator according to the proposed technical solution (figa, b) allows not only to increase the sensitivity, but also to increase its quality factor. This is due to a decrease in the resonance frequency due to a change in the shape of the bending vibrations of the resonator.

The mechanical system of FIG. 2b is equivalent to a cantilever rod of length l with a concentrated mass at the end proportional to the mass of a connecting rod of length a. The angular movement of the connecting rod θ is determined by the transverse movement Y of the articulated joint of the connecting rod with the rod l (point A of FIG. 2b). The influence of the connecting rod on the vibrations in the mechanical system of Fig.2b is manifested in the form of a force P _w acting at the connecting point of the connecting rod with the rod section of length l - point A; wherein:

where M _W - the moment due to the angular acceleration of the connecting rod;

j _w - the moment of inertia of the connecting rod relative to the axis of rotation (point D fig.2b).

For the embodiment of the connecting rod in the form of a rod with a uniform cross-section b · h, uniform in length a, and the rod, the moment of inertia is:

where m _w - the mass of the connecting rod.

- выражение (17) с учетом (18) принимает вид:Given that

- expression (17) taking into account (18) takes the form:

- эквивалентная масса шатуна.Where

- equivalent mass of the connecting rod.

и эквивалентной массы консольного стержня

и сопротивления трения S_Т.Thus, the system with a rigid connecting rod of FIG. 2b is equivalent to a cantilever rod with a concentrated mass at its free end with a value equal to 1/3 of the mass of the connecting rod (for a connecting rod with a uniform cross-section of uniform length). Its equivalent mechanical circuit can be presented in the form of parallel-connected: an elastic element with flexibility e (see figa) of two masses - the equivalent mass of the connecting rod

and the equivalent mass of the cantilever rod

and friction resistance S _T.

An analysis of the equivalent mechanical circuit of FIG. 3a is conveniently carried out using the method of electromechanical analogies. When using the first system of electromechanical analogies, a parallel connection of mechanical elements: two masses m ₁ , m ₂ , flexibility e and friction resistance S _T (see figa), is represented as a series connection of their analogues - inductances L ₁ , L ₂ , capacitor C and resistance R, respectively (see figb); the force F (t) applied to parallel connected mechanical elements is equivalent to an emf source connected in series with electrical elements - analogues. Expressions for the quality factor Q _{e of a} sequential electric resonance circuit (fig.3b) and a mechanical circuit Q _m (figa) has the form:

Where

- значения резонансных частот электрического и механического контуров соответственно,

- the values of the resonant frequencies of the electrical and mechanical circuits, respectively,

ρ _e , ρ _m - wave impedances of the electrical and mechanical circuits;

m ₁ , m ₂ are the equivalent masses of the kinematic system corresponding to the equivalent mass of the cantilever-fixed rod of length l and the equivalent mass of the connecting rod.

From the expression (21) it follows: with a constant value of compliance e of the cantileverly fixed rod and, accordingly, its resistance S _{T, the} quality factor of the system increases with increasing sum of the equivalent masses of the rod l and the rod a. The efficiency according to the proposed technical solution in terms of increasing the quality factor can be expressed through the ratio of the quality factors of two mechanical systems: a resonator with a connecting rod with an equivalent mass m ₂ and a resonator with a weightless connecting rod (m ₂ = 0), equivalent to a cantilever-fixed rod.

where m ₁ is the equivalent mass of the cantilever fixed rod.

In addition to increasing the power sensitivity and quality factor of the resonator, the proposed technical solution improves the efficiency of the electromechanical transducer of the piezoelectric type. With bending vibrations of the prototype resonator, the sign of mechanical stress from its deformation twice changes along the length of the rod. Neutral sections dividing sections of the rod with different signs of mechanical stress are at a distance of 0.224 of the total length from the attachment points of the ends of the rod. The greatest mechanical stresses are in the zone of attachment of the ends and therefore the electrodes of the electromechanical transducer of the piezoelectric type are placed in areas with maximum mechanical stresses; the length of these sections is less than 0.224 of the total length of the rod. Performing an elastic hinge in the middle part of the rod substantially changes the nature of the distribution of mechanical stresses along its length under bending vibrations of the rod; zones with one sign of mechanical stress cover most of the rod, and neutral sections will be located near the elastic hinge. This allows you to expand the area of placement of the electrodes of the piezoelectric transducer, increasing its efficiency (to reduce the equivalent resistance of the resonator).

A further increase in the efficiency of the piezoelectric type can be achieved by performing a portion of the rod 1 located between the rigidly connected end with the portion 5 of the application of the measured force and the elastic hinge 3 in the middle part, decreasing smoothly or stepwise in width (in the direction from section 5 to the hinge 3 ) The increase in efficiency with a variable width of section 1 is a consequence of alignment along the length of mechanical stresses. A sample was made that confirmed the operability of the device, the relative deviation (sensitivity) was increased by at least three times, and the quality factor was doubled.

Claims (1)

A power-sensitive resonator with a bending waveform, made at least in the form of one rod, the ends of which are rigidly connected to the areas of application of the measured force, characterized in that elastic joints are formed in the middle part and at one of the ends of the rod, the width of the rod section, concluded between the elastic hinges, more than the width of the rest of the rod portion, while the rod portions are seamlessly connected.

Images